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We present an algorithm called Improve that improves a proposed partition of a graph, taking as input a subset of vertices and returning a new subset of vertices with a smaller quotient cut score. The most powerful previously known method for improving quotient cuts, which is based on parametric flow, returns a partition whose quotient cut score is at least as small as any set contained within the proposed set. For our algorithm, we can prove a stronger guarantee: the quotient score of the set returned is nearly as small as any set in the graph with which the proposed set has a larger-than-expected intersection. The algorithm finds such a set by solving a sequence of polynomially many s -- t minimum cut problems, a sequence that cannot be cast as a single parametric flow problem. We demonstrate empirically that applying Improve to the output of various graph partitioning algorithms greatly improves the quality of cuts produced without significantly impacting the running time.